We show that \(S+T_{\in ,B}\)  does not have the amalgamation property. This suffices to prove the Fact by [1]}.
Given \((V,\in )\)  model of \(S+\text{{\sf MM}}^{++}\) , let \(G\)  be \(V\) -generic for Namba forcing at \(\aleph _2\)  and \(H\)  be \(V\) -generic for
\(\operatorname{Coll}(\omega _1,\omega _2)\) . Note that (by Thm. REF , since all predicate symbols of \(\in _B\)  are either universally Baire sets or \(\Delta _0\) -definable formulae or the non-stationary ideal on \(\omega _1\) )
\((V,\in _B^V)\sqsubseteq (V[G],\in _B^{V[G]}), (V[H],\in _B^{V[H]}).\) 
If \(W\)  is an \(\in _B\) -amalgamation of \(V[G]\)  and \(V[H]\)  over \(V\) , and \(W\)  models \(S+T_{\in ,B}\) , we get that in \(W\)  \(\omega _2^V\)  has cofinality \(\omega _1^V\)  and \(\omega \)  at the same time (both properties are expressible by \(\Delta _0\) -formulae in parameters \(\omega ,\omega _1^V,\omega _2^V,f,g\)  stating that \(f:\omega \rightarrow \omega _2^V\)  is cofinal and \(g:\omega _1\rightarrow \omega _2^V\)  is cofinal), hence \(\omega _1^V\)  is countable in \(W\) .
This is impossible since \((H_{\omega _2}^V,\in _B^V)\prec _1 (W,\in _A^W)\)  and \(\omega _1^V\)  is uncountable in \((H_{\omega _2}^V,\in _B^V)\) .
